Let z_{1} and z_{2} be two imaginary roots of | vedclass

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(C) The roots of the equation $z^{2}+pz+q=0$ are $z_{1}$ and $z_{2}$.For the points $z_{1}, z_{2}$ and the origin $(0)$ to form an equilateral triangl

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$p^{2} = 3q$

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(C) The roots of the equation $z^{2}+pz+q=0$ are $z_{1}$ and $z_{2}$.For the points $z_{1}, z_{2}$ and the origin $(0)$ to form an equilateral triangle in the Argand plane, the condition is $z_{1}^{2}+z_{2}^{2}+0^{2} = z_{1}z_{2} + z_{2}(0) + (0)z_{1}$, which simplifies to $z_{1}^{2}+z_{2}^{2} = z_{1}z_{2}$.Adding $2z_{1}z_{2}$ to both sides, we get $(z_{1}+z_{2})^{2} = 3z_{1}z_{2}$.From the properties of quadratic equations, $z_{1}+z_{2} = -p$ and $z_{1}z_{2} = q$.Substituting these values, we get $(-p)^{2} = 3q$, which implies $p^{2} = 3q$.

Let $z_{1}$ and $z_{2}$ be two imaginary roots of $z^{2}+pz+q=0$, where $p$ and $q$ are real. The points $z_{1}, z_{2}$ and the origin form an equilateral triangle if

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